Theorem grlimprop2 48572

Description: Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)

Hypotheses

Ref	Expression
grlimprop.v	⊢ 𝑉 = (Vtx‘𝐺)
grlimprop.w	⊢ 𝑊 = (Vtx‘𝐻)
grlimprop2.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
grlimprop2.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
grlimprop2.i	⊢ 𝐼 = (iEdg‘𝐺)
grlimprop2.j	⊢ 𝐽 = (iEdg‘𝐻)
grlimprop2.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
grlimprop2.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

Assertion

Ref	Expression
grlimprop2	⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Distinct variable groups:   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉   𝑓,𝐹,𝑔,𝑣   𝑓,𝐺,𝑔,𝑖,𝑥   𝑓,𝐻,𝑔,𝑖,𝑥   𝑥,𝐼   𝑥,𝐽   𝑖,𝐾   𝑖,𝐿   𝑓,𝑀,𝑔,𝑖,𝑥   𝑓,𝑁,𝑔,𝑖,𝑥   𝑣,𝑖

Allowed substitution hints:   𝐹(𝑥,𝑖)   𝐼(𝑣,𝑓,𝑔,𝑖)   𝐽(𝑣,𝑓,𝑔,𝑖)   𝐾(𝑥,𝑣,𝑓,𝑔)   𝐿(𝑥,𝑣,𝑓,𝑔)   𝑀(𝑣)   𝑁(𝑣)   𝑉(𝑥,𝑓,𝑔,𝑖)   𝑊(𝑥,𝑣,𝑓,𝑔,𝑖)

Proof of Theorem grlimprop2

Step	Hyp	Ref	Expression
1		grlimdmrel 48566	. . . . 5 ⊢ Rel dom GraphLocIso
2	1	ovrcl 7433	. . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
3		id 22	. . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
4		df-3an 1099	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) ↔ ((𝐺 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
5	2, 3, 4	sylanbrc 592	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
6		grlimprop.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
7		grlimprop.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐻)
8		grlimprop2.n	. . . 4 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
9		grlimprop2.m	. . . 4 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
10		grlimprop2.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
11		grlimprop2.j	. . . 4 ⊢ 𝐽 = (iEdg‘𝐻)
12		grlimprop2.k	. . . 4 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
13		grlimprop2.l	. . . 4 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
14	6, 7, 8, 9, 10, 11, 12, 13	isgrlim2 48569	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
15	5, 14	syl 17	. 2 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
16	15	ibi 269	1 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ⊆ wss 3904  dom cdm 5645   “ cima 5648  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  Vtxcvtx 29143  iEdgciedg 29144   ClNeighbVtx cclnbgr 48404   GraphLocIso cgrlim 48562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-1o 8432  df-map 8805  df-vtx 29145  df-iedg 29146  df-clnbgr 48405  df-isubgr 48447  df-grim 48464  df-gric 48467  df-grlim 48564

This theorem is referenced by: (None)